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Closure constraints for hyperbolic tetrahedra. (English) Zbl 1323.83011

Summary: We investigate the generalization of loop gravity’s twisted geometries to a \(q\)-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant \(\mathrm{SU}(2)\) gauge theory. Its classical states are graphs provided with algebraic data. In particular, closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in flat space \({{\mathbb{R}}^{3}}\). One then glues them, allowing for both curvature and torsion. It was recently conjectured that q-deforming the gauge group \(\mathrm{SU}(2)\) would allow us to account for a non-vanishing cosmological constant \(\varLambda \neq 0\), and in particular that deforming the loop gravity phase space with real parameter \(q\in {{\mathbb{R}}_{+}}\) would lead to a generalization of twisted geometries to a hyperbolic curvature. Following this insight, we look for generalization of the closure constraints to the hyperbolic case. In particular, we introduce two new closure constraints for hyperbolic tetrahedra. One is compact and expressed in terms of normal rotations (group elements in \(\mathrm{SU}(2)\) associated to the triangles) and the second is non-compact and expressed in terms of triangular matrices (group elements in \(\mathrm{SB}(2,\mathbb{C})\)). We show that these closure constraints both define a unique dual tetrahedron (up to global translations on the three-dimensional one-sheet hyperboloid) and are thus ultimately equivalent.

MSC:

83C45 Quantization of the gravitational field
53Z05 Applications of differential geometry to physics
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
57M50 General geometric structures on low-dimensional manifolds
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